Lec 11 | MIT 18.01 Single Variable Calculus, Fall 2007

The professor begins by acknowledging the support provided under a Creative Commons license, which helps MIT OpenCourseWare offer free educational resources. Donations can be made at ocw.mit.edu. The lecture is a continuation of the topic on sketching functions, with a particular focus on hyperbolic functions. The professor admits to being slightly behind schedule but aims to catch up by the following Tuesday. The lecture proceeds with an example of sketching the function f(x) = x + 1 / (x + 2), emphasizing the importance of not giving up even when the derivative does not equal zero, indicating no critical points.

The professor elaborates on the function from the previous lecture, discussing the importance of recognizing where the function is undefined, such as at x = -2. The behavior of the function as x approaches this value from both sides is analyzed, revealing the function tends towards negative infinity on one side and positive infinity on the other. The professor also discusses the behavior of the function at the 'ends' (as x approaches plus or minus infinity) and uses limits to show the function approaches y = 1. Asymptotes are introduced to sketch the function's graph, and the professor highlights the importance of considering the function's behavior at these critical points to avoid errors in graphing.

The professor continues with the function sketching example, addressing the concern of whether the function might dip below a certain point. A student correctly points out that since there are no critical points, the function cannot double back on itself. The professor emphasizes that the derivative being non-zero ensures the function will not backtrack. The discussion then turns to the importance of recognizing the function's behavior at discontinuities and ends, and the professor outlines a general strategy for sketching functions, which includes plotting discontinuities, endpoints, and easy points as a pre-calculus approach before considering derivatives.

The professor outlines a general strategy for sketching functions, which includes four main steps: plotting discontinuities and endpoints, determining the sign of the first derivative to establish increasing and decreasing intervals, examining the second derivative to determine concavity and inflection points, and combining all information to sketch the function. The strategy is illustrated using the function f(x) = x / ln(x) as an example. The professor emphasizes the importance of considering the function's behavior at the endpoints and discontinuities, as well as the role of the first and second derivatives in sketching the function.

The professor discusses the process of finding critical points and analyzing concavity in the context of function sketching. Using the function f(x) = x / ln(x), the professor demonstrates how to find where the derivative is zero and how to determine the intervals where the function is increasing or decreasing. The second derivative is introduced to analyze the function's concavity and to identify inflection points. The professor also advises against computing the second derivative unless necessary, as it can be complex and time-consuming.

The professor continues the example of sketching the function f(x) = x / ln(x), focusing on identifying vertical asymptotes and endpoints. The function's behavior as x approaches 1 from both sides and as x approaches infinity is analyzed, revealing a vertical asymptote at x = 1 and the function's behavior at the endpoints. The professor emphasizes the importance of these points in understanding the overall shape of the function's graph.

The professor provides a detailed analysis of the function f(x) = x / ln(x), identifying critical points and analyzing the function's concavity. The derivative is computed, and the professor demonstrates how to find where the derivative is zero, leading to the identification of a critical point at x = e. The function's behavior in different intervals is discussed, and the professor uses the second derivative to analyze the function's concavity, revealing an inflection point at x = e^2.

The professor concludes the lecture by discussing the graphical representation of maxima and minima in the context of function sketching. The importance of considering critical points, endpoints, and points of discontinuity when identifying extrema is emphasized. The professor also hints at the goal of finding shortcuts to determine maxima and minima without the need for extensive graph decoration or detailed sketching, setting the stage for future lectures on the topic.